A bit more complex than that, but yeah. Thatcher brought the hammer down and provided the Argentinean government at the time with the distraction it desperately wanted (due to anti-government protests).
I'm not sure what honestly irks me more about it. The fact she played right into the junta's hands with her response, or that she was so rapidly prepared to shed blood for territories that had never been more than a footnote. Hell, the only reason Britain retained sovereignty into the 80s to begin with is because nobody thought them significant enough to detach.
I suspect that the latter was more of a driving factor -- there was a general crossing-party-lines sense that the UK had lost enough territory after 1945, thank you (even
Spitting Image, no friend of the Tory government
, played into that with their cheeky anthem rewrite "Rule Britannia/Britannia rules not much!"), and the Foreign Office was telling the PM at around the same time that they should let China take back not only the leased territories of Hong Kong in 1997, but also Hong Kong Island proper (which had been ceded in perpetuity to the UK by the Qing dynasty after the First Anglo-Chinese War). Given that the Communist Party never recognized the treaties signed in the imperial era, and the far less certain outcome of a Third Anglo-Chinese War, the war that Thatcher got was probably a better outcome than the war she wanted.
One other side effect for the UK -- the still newly-democratizing Spain (King Juan Carlos ordered the army back to its barracks in the same year when some officers decided the experiment had gone on long enough) stopped asking for Gibraltar back after the Falklands War, and to my knowledge even the Brexit chaos and dragging out of negotiations hasn't brought up that bit of land. It's even more of a rock (at least you can raise sheep on the Falklands), but the UK has its naval base, for whatever good that'll do now.